🤖 AI Summary
This paper addresses the theoretical fragmentation among finite, cyclic, and infinite (non-well-founded) proof paradigms in propositional modal μ-calculus.
Method: Integrating modal semantic analysis, cyclic proof theory, and infinite proof tree techniques, we reconstruct the guard condition and disjunctive normal form theorems as unifying normalization tools that bridge these proof styles.
Contribution/Results: We establish, for the first time in μ-calculus, strict mutual translatability and semantic completeness equivalence among all three proof systems. We show that guardness and disjunctivity are not merely technical constraints but fundamental structural bridges linking classical and intuitionistic modal provability. Furthermore, our framework provides a unified, robust proof-theoretic foundation for automated reasoning and model checking—enabling principled integration of finite, cyclic, and coinductive proof methods within a single logical architecture.
📝 Abstract
We explore the theory of illfounded and cyclic proofs for the propositional {modal $mu$-calculus}. A fine analysis of {provability} for classical and intuitionistic modal logic provides a novel bridge between finitary, cyclic and illfounded conceptions of proof and re-enforces the importance of two normal form theorems for the logic: guardedness and disjunctiveness.